Internat. J. Math. & Math. Sci.
VOL. 18 NO. 3 (1995) 551-560
551
TRANSCENDENTALITY OF ZEROS OF HIGHER DERIVATIVES OF FUNCTIONS
INVOLVING BESSEL FUNCTIONS
LEE LORCH and MARTIN E. MULDOON
Department of Mathematics and Statistics
York University
North York, Ontario
Canada M3J 1P3
(Received September 2, 1995 and in revised form February 2, 1994)
ABSTRACT. C.L. Siegel established in 1929 [Ges. Abh., v.1, pp. 209-266] the deep results that
(i) all zeros of J(z) and J’(z) are transcendental when , is rational, z :p 0, and (ii)
J’,,(z)/J(x) is transcendental when , is rational and z algebraic. As usual, J(z) is the
Bessel function of first kind and order t,. Here it is shown that simple arguments permit one to
infer from Siegel’s results analogous but not identical properties of the zeros of higher derivatives
of x-UJ,(z) when / is algebraic and v rational. In particular, J"(4-x/)= 0 while all
other zeros of J"(z) and all zeros of J’,,"(z), v # 1, z # 0, are transcendental. Further,
Jo’)(+x/) 0 while all other zeros of Jo’)(z), z =p O, and of J’)(z), v # O,z y O, are
transcendental. All zeros of J")(z), z O, are transcendental, n 5,..., 18, when v
is rational. For most values of n, the proofs used the symbolic computation package Maple V
(Release 1).
KEY WORDS AND PHRASES. Bessel functions, zeros, transcendentality, differential equations.
1991 AMS SUBJECT CLASSIFICATION CODES. 33A40, llJ81
1.
INTRODUCTION.
Carl Ludwig Siegel established [3; 2, Ch. 6, 2, p. 217] the deep results that when v is
rational the zeros of J,(x),J(x),x # O, are transcendental and that the ratio J(x)/J,(x) is
transcendental for x algebraic. Here J,,(x) is, as usual, the Bessel function of the first kind
and order v.
From these properties, as will be shown below, analogous (but not always identical) results
can be inferred by quite simple arguments for the zeros of higher derivatives of
3"..(.)
z-"J().
when # is algebraic and u rational.
For / 0, this gives full information on the transcendentality of zeros of J(")(x),n
t,
m + 7, m
0,+1,+2,..., the
2,...,18. Other standard special cases are t
7,
familiar spherical Bessel functions (a multiplicative constant aside), and / t which presents
the standard entire function A(x), again disregarding a multiphcative constant.
552
L. LORCH AND M. E. MULDOON
The transition from Siegel’s theoreln to these results is via minor manipulations of the
differential equation for y J,,(z) [5, 4.3a(9), p. 98]
x2y" + (2# + 1)xy’ + (x
-
+ l:)y
v,2
O
(1)
These calculations can be applied equally well to the general form
xy + p(x)xy’ + q2(x)y
O,
(2)
where p2(z) and q:(x) are algebraicpolynomials.
For y satisfying (2), it follows by induction that y also satisfies, for n
"y(") + ,.,():y’ + q.(:)y
2,3,4,...,
o,
(3)
where p,(x), q,(x) are algebraic polynomials. The subscript n does not suggest the degrees
of these polynomials. (In fact, deg(p,) 2[(n 1)/2] and deg(q,) 2[n/2].) It records only
their association with the differential equation (3) of order n.
The induction from order n to order n + 1 leads to
"+U "+’ + [p’.() +
p,(){1 -. p()} + q-()]U’
nq,(x) q2(x)p,(x)]y 0,
(4)
+ [xq(x)
where the coefficient functions are again polynomials. The passage is achieved by differentiating
(3), multiplying through by x, then replacing x"y(") from (3) and x’y" from (2). From (4),
we have
p,+,
xp(x) + p,(x){1
q,,+l
n
xq’,,(x)- nq,,(x)
p2(x)} + q,.,(x),
(5)
q2(x)p,.,(x).
(6)
Thus, (3) and (4) express the higher derivatives of ,]’,u(x) in terms of ,Y’,u(x) and
.L’,.().
This will permit the application of Siegel’s theorems to
J’,.()
,.7,,,u(x)
is
s’()
J,,(x)
.7(")(x),
since
(7)
x
Hence, ,’ (x)/,Y’,,,u(x) is transcendental whenever x(# 0) and / are algebraic and v
rational, since this is the case for J’(x)/J,,(x). One observation is immediate from Siegel’s
work and does not require an appeal to the differential equation:
THEOREM 1.
,7’,u (z), x
If u is rational and
transcendental.
is
0,
#
PROOF. Let
#0
be any arbitrary zero of
/
algebraic,
then
any
zero
of
,7,u(x ). From (7),
J()
were algebraic, then this equation would imply that J’,,()/J,,() is algebraic. But [2,
If
Ch.6, 2, p.217] this is false when v is rational as here.
Proofs of the remaining results will use special cases of (3). The next theorem, however, will
be phrased more generally so as to make it potentially applicable to yet other cases.
THEOREM 2. Let y(x) be a non-trivial solution of the differential equation (3) where
p,.,(x), q,,(x) are algebraic polynomials. Suppose for each algebraic x # 0, that (i) y(x) # 0
ZEROS OF DERIVATIVES OF FUNCTIONS INVOLVING BESSEL FUNCTIONS
and that (ii) y’(x)/y(x) is transcendental. Then a zero
and only if p,(() q,(() 0.
0 of
553
y(")(x)
is algebraic if
PROOF It is obvious that # 0 is an algebraic zero of y(")(x) if p,() q,() 0.
What remains, then, is to show that y(")()
0, ( 0, implies p,()
q,(() 0 when
( is algebraic
With
# 0 algebraic, it is immediate from hypotheses (i) and (ii) that y’() # 0. Fro,n
(3), (p,()y’(() + q,()y() 0. If p,(() # 0, then
:
q,,()
y’()
()
The right hand side is algebraic for
#
0 algebraic but, by hypothesis, the left hand side is
not.
Hence p,() 0 and so q,()y()
(i), and the proof is complete.
0. Furthermore, q,() also vanishes, according to
Thus, the algebraic zeros other than 0, if any, are the common roots of p,(x) and q.(z).
The search for such can be made via various methods of elimination, going back, say, to Euler,
Bdzout and Sylvester.
An apphcation of Theorem 2 to ff,,(z), into which Theorem 1 is incorporated, can be
formulated as follows:
COROLLARY. If v is rational and # algebraic, then all but a finite number (perhaps
none) of the infinitely many zeros of .?(")(x) are transcendental, x :/: 0, n 0,1,2,
The remaining results use the specific differential equations (1) and (4).
THEOREM 3 If t, is rational and # is algebraic, then any zero # 0 of fl.";,gt (z)
gl
is transcendental unless #
algebraic zeros,
+#
-
0 where
2
and v
v2_
#
l_
.
In the latter
cases there are precisely two
4"
REMARK. Note that these algebraic zeros are imaginary when
COROLLARY. An zeros, other than 0, of
scendental when v is rational.
v
< 1/4.
d’,,’(z), [d,,(z)/z1/2]" and [d,,(:r)/z"]"
are tran-
PROOF. Applying Theorem 2 to differential equation (1) shows that a zero
# 0 of
is
when
and
when
and
1
v2
only
algebraic
p()
q2()
,.7’,’t,(z
2# +
+ #2 vanish
2
fi)r
sinz and
/-dt(z)
of
second
their
derivatives
are all
respective
0,
vJ_ }(z)
non-zero rational multiples of r and hence transcendental.) The Corollary is immediate.
.q’’ (x non-zero algebraic zeros arise even when #
For ..,,,,,
0 i.e for J’"(x). Specifically,
when
other
1.
are
o
transcendental
All
zeros, of J’"(z)
J’"(+CS)
for any rational ,
including :l:l.
simultaneously. This estabhshes Theorem 3.
J/Tr
(It may be noted that
cos z. The zeros, other than
If v is rational and # algebraic, each zero
THEOREM 4
is
# 0 of .q’"(z)
t,t
transcendental except when t
with
I,
#2+(#+ 1)(2# + I) 4# 3+9# +5# +
2
(# + 1)(2# + 1)(2# + 3) and with (4# 3 + 9# + 5# + 1) rational.
COROLLARY. For
when v
when v2
[J,(z)/:r 4-1
1 with
-i-2s
with
are
2
2
v
3,
rational, (i)any
zero
#
(ii) any zero
12, and (iii) all
#
0 of
zeros
0 of
J; (x)
[J(x)/x1/2]’"
(other than zero)
transcendental without exception.
is transcendental except
is transcendental except
of [J.(x)/x+"] and of
L. LORCH AND M. E. MULDOON
554
PROOF. When
2 and y
n
’"
+ [-
J,,,u(x), the differential equation (4) becomes
2(, + 1)(2 + 1)]u’
+
[(2t + 1)x + (2 + 3)(. u)]y 0.
(8)
The coefficients of y and x’ vanish simultaneously when and only when u
+(+
with
x
1)(2 + 1)
( + 1)(2 + 1)(2 + 3), as asserted in th theorem. In adtion, since
u is rational, so too must be [ + ( + 1)(2 + 1)].
This establishes the theorem. Parts (i) and (ii) of the corollary are immediate. When
the foregoing shows that coecients of y and xy’ vanish only when
u
-1,sothat x 0. This verifies the assertion in (iii) when
iu. If
19,
1, then u
so that u cannot be rational as required. If
0. Together these last two
-1, then z
sentences verify the remaining assertion in part (iii) of the corollary.
The well-known relation Ja(x) -J(x) makes it obvious that J4)(i) -J"(i)
0, so that J4)(x) possesses a pair of gebraic zeros (other than zero) for a specific rational u, namely v
0. It wilt be seen that all other zeros (other than zero) of J)(x) are
transcendental when v is rational, including those for which v O, x # 3.
This will emerge from the following theorem.
,
>
THEOREM 5.
If u is rational and p algebraic, then any zero
transcendental except when
0 of
JJ)(x)is
.-(32."(.++60.x)[.+ +284.1. ++ z204. + 49)]
with
:
-(2p
+ 3)[-4#
12p
7 + (32#
(9)
+ 160# 3 + 284p + 204it + 49)]
(10)
and with
{p
(2/ + 1)[(4# + 12 + 7) :k (32# + 160# 3 + 284# + 204/z + 49)]}
1/2
(11)
rational.
COROLLARY. For
0 of JJ4)(x)is transcendentMexcept
rational, (i each zero
when
0 th ( 3, (ii) each zero ( # 0 of [J(x)/x]]() is transcendentM and (iii)
each zero
0 of [J(x)/x] (4) is transcendental except when
0 with
3.
PROOF. When
coefficients of xy
n
and y
p4(x)
-2(2p + l)z
and
q4(x)
-
4, and th A
--x
p-u
,
the differentiM equation (3) has as respective
2(2p + 3)A + 2(p + 1)(2p + 1)(2p + 3)
+ [(2 + 1)(2 + 3)- 2A]x + (4 + 12 + 11)A
A 2.
These expressions are calcated by applng (4) with n 3 to (8).
The algebraic zeros of .7()(z) arise om the common zeros of p4(x) and q4(x), x # 0.
be
When p =-, p4
onlywhen A=0 so tht in this cse q4(z) =-z
wi .ise o.y w.
0. Z.s,
.o gei eo ote
2()
To determine the common zeros of
than perhaps z 0. Henceforth, we consider
#
o wm V,() 0 i ,tit,tea i,
W() ,a q,(),
#
# 0, te a,eo
q4(z). On simpfication, q4(z) 0 then becomes
..
-,
4
z(2. + 1)(4. + lZ + z)
-,
-.
.(. + 1)(z + )(2u + )
o.
()
ZEROS OF DERfVATIVES OF FUNCTIONS INVOLVING BESSEL FUNCTIONS
555
Since A l2-v 2, the solutions of the quadratic equation (12) yield (9). The values (10) result
from replacing in the equation p4(.r) 0 the expression for v given by (9). The restriction
(11) merely restates the hypothesis that v is rational. This proves the theorem. The assertions
in the corollary are easy consequences in light of (12).
Remark. Condition (11), like the corresponding restriction in Theorem 3 implies the nonexistence of algebraic zeros in ,nan3’ cases. For example, if p.
1.2, 3, these requirements are
not ,net and so neither .7""’ (z) ,,or .74)(z) have algebraic zeros when /
1 2,3, x # 0.
2. RESULTANTS AND MAPLE V.
Even in the special case/ 0, the polynomials p,,, q, become substantially ,nore complicated
as n increases. For example:
p5(x)
-x
+ (2u + 7)x
u
35u
24,
and
qs(x)
--
p6(x)
and
q(x)p(x)
and
qr(x)
x
x
2Ix -6(u + l)x + 5u2(u + 5)],
3Ix (6t2 + ll)x + 5u + 75u + 40],
3( + 3), + 3(u + 26u + 20)z
3( + 5)x + 3(u
us-85u
+ 50 + 64)x 2- us- 175u
274 4,
1624t
720,
3[-x s + (9r, + 17x *) -(15r, + 190r? + 120)x + 7r, s + 245t, + 588r,].
The method of {}1 is, in principle, still applicable. In the case n
ps(z) and as(z) have a common zero if, and only if,
8, for example, we find that
-31907119104v 30 847042707456v 2s
-4180678fi80576v 2 614494258827264v 2’t
-4099235438453760v 22 16797585600728640v 2
-446224810865120v TM 74605419864103690v s
-52227966835676760v’" + 89924792224537360v
+3400114fi3648979674v + 547044103885835061v s
+ 549201731824654896v s + 359020825792238369v
+144928821192067000v + 3040143522850000 0.
’
’
This equation has precisely one positive root. It is located between 1.25 and 1.3, the polynomial
1.25 and negative when v
1.3. For the root to be rational
being positive when v
(expressed in lowest terms), its numerator must be a factor of the constant term
24(3 )(7s)(533),
3040143522850000
its denominator a factor of the leading coefficient
31907119104
2-(32)(72)(23).
show by checking that no such rational root is possible, so J(S)(z) has no algebraic
zero, except possibly z 0, for v rational.
Rather than giving more and more complex ad hoc arguments for larger values of n, we will
use the concept of resultant. The resultant (or ehminant) [4] of two polynomials p and q is a
One
can
L. LORCH AND N. E. NULDOON
556
function r of the coefficients which vanishes if and only if the polynomials have a common root.
The resultant may be expressed as a determinant involving the coefficients or it inay be found
[6] using the Euclidean (greatest common divisor) algorithm. The proofs of Theorems 3, 4 and
5 may be rearranged by finding the values of p and u for which resultant r,,(tL u) of p, and q,
vanishes. We then substitute these values back into p, and q, and check to see whether they
have a common non-zero root, i.e., whether their g.c.d, is anything other than a power of x.
For example, in Theorem 3, the resultant of p2(x) and q2(x) is 2p + 1 which vanishes for
-1/2, ,v get p2(x) 0 and q2(x) x v + 1/4 which have a common
-1/2. Fo,
non-zero root
For
n
only if
u
.
#
_> 5, we concentrate on the special cases
0, 1/2, u and -1/2.
#
2.1 The case/ 0. We used Maple V (Release 1) to form the polynomials p, and q,, and
to set up [1, pp.178-179] and factor their resultant r, a polynomial in v. Here is the procedure
used on a SUN 4/75 at York University.
p
proc (n)
p(1) := !"1,1];
19(2) :=[2.m+1, x**2-nu’2+m’2]
if n <= 2 then RETURN (p (n) fi;
for i from 2 to n-I do
p(i+l) :=
[x*diff(p(i)[l], x) + p(i)[l]*(l-i-p(2)[l]) +p(i)[2],
x*diff(p(i) [2] ,x) -i*p(i) [2] -p(2) [2]*p(i) [1]]
od;
RETURN (p (n)
r
rr
end;
proc(n) resultant(p(n) [1] ,p(n) [2] ,x) end
proc(n) factor (r (n) end
The following expressions were obtained for the first eight resultants factored over the rationals.
rl(v)
1, r2(v)= 1,
,
1)2(u + 1)
r4(u) 64v(7 + 2u2) 2,
r3(u)
rs(u)
r6(v)
4(v
16(877u + 493u + 448u + 432)2,
11664(-10000
-47208u
rT(U)
12576vs
746496(10800u 14 + 1045744v ’2
103853517u e
rs(u)
16723u
48749v 4
+ 256u’) 2,
3311544v
’
51138608u 8
105987611u -48750264u
-
12150000) 2,
47775744(18464768u is + 351700992u 16 + 20996299776u 14
/ 185371620544u
747805126038u 6
+ 278668777744u
1012303674997u
72563241789u 8
626716971000u:
204205050000)
.
ZEROS OF DERIVATIVES OF FUNCTIONS INVOLVING BESSEL FUNCTIONS
557
Hence
v
+1 arc the only (rational) roots of r3(v). This is consistent with part (i) of the
18 and there are no
Corollary to Theorem 4. We carried out the computation as far as n
,s
further rational roots of r,(u). Hence, for these values of n, there no possibility of a common
root of p, and q,, and hence no non-zero algebvazc root of J(")(x), when u is rational. Our
results for
0 can be summarized as follows:
THEOREM 6. If // is rational and x :p 0, then all zeros of J"l(x), n 0,1,2,...,18
arc transcendentaI’with the exception of the zeros +V which occur in the cases n
3, v +1
andn=4, //=0.
2.2. The case #
1/2. We used Maple V again to set up the polynomials p, and q, and
their resultant r,, factored over the rationals. The first few results are as follows:
1, r(u)= 4,
rl(u)
1
(2//- 5)2(2//- 5) 2,
r3(/)
r4(u)
vs(u)
r6(u)
rT(//)
247 +
16//4)
, ,
4(36324u + 9909 + 3056u + 1216u6)
(729/4)(8u + 13)2(256u s --29952u 6- 155808u
(729/1024)(1105200v TM + 2105987072v I
1878589556672v
rs(u)
(216v
3941745616400v
16848v
--845743)
,
31126270976v l -4105477640v s
3183516258012v
3944054162017)
,
729(5754585088v TM + 146218352640//’6 + 24293948915712v TM
+ 179297378451456u ’2
31107275263623680u
628056658616320u
812967841220864v s
71858687471992512u -82804731870624480v
55560127536055125) 2.
The calculations were carried as far as n 18. The only case where a rational root exists is
the case n 3, u
25/4; the corresponding roots of [x-’/2J(x)] (a) are at 4-Vfi-. This agrees
with part (ii) of the Corollary to Theorem 4.
We summarize our results as follows:
THEOREM 7. If v is rational and x # 0, then all zeros of [z-x/2J(x)] ("), n
0,1,..., 18 are transcendental with the exception of the roots 4-v/ occurring for n 3, v
25/.
u. In contrast to the situation encountered in Theorems 6 and 7, here
2.3. The case tt
the functions r,(u) appear to have rational roots for every n > 2. So we have to check further
to see if these rational roots correspond to non-zero common roots of p,(x) and q,(x). We
again have recourse to Maple V, first to evaluate the roots of r,(v), then to find the greatest
common divisor (using god [1, pp. 100-101])of the polynomials p,(x) and q,(z) in the special
cases when these this roots are substituted for v. The Maple procedure of 2.1 was augmented
as follows:
558
L. LORCH AND M. E. MULDOON
1. oo 1
gcd
2
3
-1/2
-1/2
4
-1
0
x
x
x
-3/2
1
5
-1
x
-3/2
x:
-2
x
x
x
x
x
z
x6
x
-3/2
-2
-5/2
-1/2
-1
-3/2
Z
-2
x
x
-5/2
-3
-1
--2
-3
9
-5/2
-7/2
-1/2
--3
-7/2
-1
-4
-3/2
-9/2
-2
--5
-5/2
12
-3
-1/2
--1
-7/2
-3/2
-4
10
-1/2
-3/2
--5/2
--2
-1/2
-5/2
-1
--3
-3/2
-7/2
-2
-4
-5/2
-9/2
-3
--5
-7/2
-11/2
--4
x:
-9/2
Table 1" Case #
rrr := proc(n) root s (r (n)
11
-2
15
x
-1
7
x:
-1/2
-1/2
6
-1
x
x
x
-1
-1/2
-3/2
3
x
-1/2
l rt lgcdl
8
v
end
rrrr := proc(n)
nops(rrr(n)) do
a := subs(m
op(1,op(i,rrr(n))),p(n) [1]);
b := subs(m
op(1,op(i,rrr(n))) ,p(n) [2])
c := gcd(a,b);
print (n,i ,op(1, op(i, rrr (n)) ), c)
for i to
od
end
Table 1 gives the output of this procedure in the case/ v.
The left column contains values of n. The second contains the rational values of v for which
the resultant of p, and q, vanish and the third column contains the corresponding gcd’s for p,,
and q,. The only gcd’s with a root other than 0 in the table correspond to the cases n 4, v 0
and z
3 (see part (i) of the Corollary to Theorem 5) and n
15. We
5, v 1 and x
have carried the table to n 13 without encountering any other exceptional cases. This proves
the following:
THEOREM 8. If v is rational and z # 0, then all zeros of [z-,’J,,(z)]O}, n 1,..., 13
are transcendenta! except for zeros at + / in case n 4 and v 0 and at 4- in case n 5
and v= 1.
The Table and its continuation suggest a number of conjectures:
ZEROS OF DERIVATIVES OF FUNCTIONS INVOLVING BESSEL FUNCTIONS
(a) That ,’,,(v)
is a polynonfial of degree
(b) That r,,(v) has exactly
,
n(n- 1)in
559
u.
rational roots in case n
y 4,5 and n
rational roots in case
n- 4,5.
(c) That the rational roots of r,(v)
case n
4 and
2.4. The case IL
in case n
-1/2. In
are
-m/2, m
1,..., n
plus the additional root 0 in
5.
this case, we get
() =0,
r3(u)
(2u- 1)2(2u + 1) 2,
r4(u)- (2u- 1)4(2u + 1)4,
rs(v)
r6(v)
r(v)
(18225/I024)(2v
rs(U)
324(2v- 1)(2v + 1)(4v + 1) 2,
729(2v- I)6(2v + i)6(-224v
i)6(2v + i)6(20v
,
+ 16v 4- 457)
29)(432v + 16200v 7277)2(4v + 17)
,
.
1)s(2u + l)s(Z606460448u + 4463010256u + 4314561883
346449152u 6 + 22256384u s + 1605632u’)
59049(2u
What appears to be happening here is that, for each n _> 3, the polynomial rn(v) has exactly
two rational zeros and -1/2, but these never give rise to common zeros of p, and q,. This has
been verified for n 3,..., 15 using a procedure similar to that used in the proof of Theorem
8. Taking into account the special case arising in Theorem 3, we have:
THEOREM 9. If v is rationaland z :# 0, then all zeros of [z’/2J(z)]("),
are transcendental, except for zeros at 41/4 in case n 2 and v y g.
3. CLOSING REMARKS.
The results reported here suggest the following two questions
n
1,..., 15,
(for which we expect negative
answers):
1. Is there any pair of values v,n, where v is rational and n
J")(z)
has an algebraic zero other than z
19,20,..., such that
07
2. Is there any pair of values v,n, where v is rational and n
19,20,..., for which
there is more than one algebraic
0 such that J(")(4-) 07
Theorems 7, 8 and 9 suggest analogous questions and conjectures for the zeros of [z/J,.,(z)]O),
[z-"J,,(z)](") and [:r-a/2J,,(z)]O"), respectively.
The degrees of the polynomials p,, q, corresponding to J,,.u(x), p # --] can be determined precisely: They are deg (p,)
2[(n- 1)/2] and deg(q,) 2[n/2], where [-] denotes
the greatest integer function. These values provide upper bounds for the numbers of possible
algebraic zeros of ,.7,,(,u)(:r), but presumably quite wide of the mark.
ACKNOWLEDGMENTS. The authors’ research was supported by the Natural Sciences and
Engineering Research Council of Canada. We are indebted to Jason Brown of York University
560
L. LORCH AND M. E. MULDOON
and Anders K/illstr6m of Uppsala University for performing symbolic algebraic calculations for
an earlier version of this paper. The success of these efforts led us to the more systematic use
of Maple reported in 2.
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